Simulations of Quantum Dimer Models

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1 Simulations of Quantum Dimer Models Didier Poilblanc Laboratoire de Physique Théorique CNRS & Université de Toulouse 1

2 A wide range of applications Disordered frustrated quantum magnets Correlated fermions on frustrated lattices (Mott) XXZ (frustrated) magnets under magnetic field Ultra-cold atom systems Josephson junction arrays Spin orbital models Quantum computing 2

3 OUTLINE Quantum dimer models for frustrated magnets From short-range valence-bond basis to dimer basis... Rokhsar-Kivelson expansion Green function s QMC for QDM s: a new phase Doped quantum dimer models Spinon doping: still a (non-frustrated) bosonic problem Holon doping: minus sign problem! 3

4 4

5 5

6 6

7 How to deal with overlaps? SU(2) Valence Bond dimer covering Orthogonal basis by construction - Sutherland, 1988: Length of the loops of overlap graph Loop formula also for matrix elements of S=1/2 Heisenberg hamiltonian 7

8 Systematic loop expansion Mambrini, Ralko et al. (unpublished) Order 2... Order 4 Order 6 & 8!!... 8

9 Keep only plaquette terms Effective model Quantum Dimer Model Rokhsar & Kivelson, PRL 88 A typical (hard-core) dimer covering of the square lattice Dimer flip 9

10 QDM can also describe (hardcore) fermions/bosons on frustrated lattices at fractional filling U= and V >> t extended-hubbard model Quantum Loop or Quantum Dimer model insulator! Ice rule constraint: Exactly 2 or 1 particles/tetraedra Pollman, Fulde et al. DP, K. Penc & N. Shannon, PRB 75, (2007) DP, PRB 76, (2007) F. Trousselet, DP & R. Moessner, ArXiv

11 Belong to the class of constrained models Hilbert space cannot be written as tensor product of HS of sub-systems! Emerging global conservation laws Interesting topological properties 11

12 Rokhsar-Kivelson point For J=V: sum of projectors exact GS with energy E=0 Infinite-T Classical DM 12

13 Short-range RVB liquid on the triangular lattice Moessner & Sondhi, PRL 2001 Finite correlation length Exponential decay of dimer-dimer correlations E Degeneracy from topological order GS have different winding numbers { 13

14 Rich phase diagrams at zero doping: valence bond crystals and RVB phases Ralko, Poilblanc & Moesner, PRL (2008) Moesner & Sondhi PRB (2001) Syljuasen, PRB (2006) 14

15 Symmetry analysis Lanczos exact diagonalizations 8x8 15

16 Green s function QMC (T=0) Define series: used as probability for stochastic Markow process STATISTICAL SAMPLING 16

17 GFMC dynamical quantities Up to 22 x 22 lattices 17

18 Structure factors 18

19 Phase diagram Columnar Plaquette Mixed 19

20 Motion of monomers Inject monomers (by pairs) & add new term to H: 20

21 «Monomer» doping Spinon doping (magnetic field) Ralko, Becca & DP, PRL (2008) Holon doping (chemical substitution) DP, PRL 100, (2007) 21

22 Spinons get fully polarized... Spinon are polarized («Nagaoka effect») Exact diagonalisations of small clusters 22

23 Maxwell construction for phase separation Square lattice V/J=0.9 & t/j=0.1 (GFQMC) J, t > 0 : No - sign problem Phase separation doping ratio A. Ralko et al., PRL (2007) 23

24 Phase diagram vs spinon doping / magnetic field square lattice J, t > 0 : No - sign problem GF Monte Carlo square lattice triangular lattice VBC insulator boson condensation of spinons! A. Ralko, F. Mila & DP, PRL (2007); A. Ralko, F. Becca, DP, PRL (2008) 24

25 Holon doping The GS wavefunction acquires nodes! A doped QDM with a minus sign problem 25

26 holon vs spinon doping? monomers definition of dimer operator «bare» monomer statistics Sign of J «true» monomer statistics Chemical doping holons bosonic J<0 non- - Frobenius bosonic/ fermionic Applied Magnetic field spinons bosonic J>0 Frobenius bosonic Input in Hamiltonian emerging 26

27 PRB 89 Variational RVB Role of topological defects: vortex or Z 2 vison Holon (boson) + vortex spinless fermion! 27

28 Holon doped QDM Physical case: J<0 QMC replaced by Lanczos Exact Diagonalizations Holon (boson) + «vison» 6 holes/ 6x6 spinless fermion 28

29 2-holon bound-state in d-wave channel 2-holon propagator string picture 29

30 Summary / Conclusions Can replace efficiently microscopic models since easier to simulate To do list: for frustrated magnets, carry out systematic loop expansion => more complicated QDM s but minus-sign problem can still be avoided! doped systems: optics, transport, impurity doping, connection with Z 2 gauge field theory 30

31 Collaborators Arnaud Ralko (Post-doc -> Institut Néel, Grenoble) Fabien Trousselet (PhD) Roderich Moessner (Dresden) Matthieu Mambrini (Toulouse) Karlo Penc (Budapest) Nic Shannon (Bristol) Federico Becca (Trieste) Frédéric Mila (Lausanne) 31

32 Pyrochlore lattice Checkerboard lattice A lattice for theorists! 32

33 Phase diagram vs doping Role of gauge field Flux quantization: in terms of h/2e 2e-Cooper pairs VBC insulator 0-flux 1/2-flux quantum 33

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